For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$. Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$. Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.
For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$.
Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$.
Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.